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Prof Andrea Milani

Andrea Milani is Full Professor of Mathematical Physics in the Department of Mathematics of the University of Pisa and President of the company SpaceDyS. His areas of research include the N-body problem, stability of the Solar System, asteroid dynamics and families, satellite geodesy, planetary exploration, orbit determination, and asteroid impact risk. He is author of 5 books and more than 120 research papers published in peer reviewed journals.
He is the founder of the two web services, NeoDyS and AstDyS, dedicated to providing general information on all known asteroids, and the developer, together with his collaborators over the years, of the software suite ORBFIT.
He is President of the IAU Cross-Division Commission X2 (Solar System Ephemerides) and  has served as president of the IAU Commission 7 (Celestial Mechanics and Dynamical Astronomy). In 2010 he was awarded the Brouwer Award by the AAS Division on Dynamical Astronomy.

Chaotic Orbit Determination and shadowing lemma
Orbit determination is possible for a chaotic orbit of a dynamical system, given a finite set of observations, provided the initial conditions are at the central time. We test both the convergence of the orbit determination procedure and the behavior of the uncertainties as a function of the maximum number n of map iterations observed; this by using a simple discrete model, namely the standard map.
Two problems appear: first, the orbit determination is made impossible by numerical instability beyond a computability horizon, which can be approximately predicted by a simple formula containing the Lyapounov time and the relative round-off error. Before the computability horizon the least squares solution is a finite-time analog of a shadowing orbit for the observed state.
Second, the uncertainty of the results is sharply increased if a dynamical parameter (contained in the standard map formula) is added to the initial conditions as parameter to be estimated. In particular the uncertainty of the dynamical parameter, and of at least one of the initial conditions, decreases like na with a < 0 but not large (of the order of unity). Again, this is a finite-time analog to the shadowing lemma. If only the initial conditions are estimated, their uncertainty decreases exponentially with n, thus it becomes very small.
All these phenomena occur when the chosen initial conditions belong to a chaotic orbit (as shown by one of the well known Lyapounov indicators). If they belong to a non-chaotic orbit the computational horizon is much larger, if it exists at all, and the decrease of the uncertainty appears to be polynomial in all parameters, like na with a approximately 1/2; the difference between the case with and without dynamical parameter estimated disappears. These phenomena, which we have investigated in a simple model, have significant implications in practical problems of orbit determination involving chaotic phenomena, such as the chaotic rotation state of a celestial body, the chaotic orbit of a planet-crossing asteroid undergoing many planetary close approaches, and the orbit of a spacecraft performing a satellite tour with many satellite close approaches.

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